Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 52 / 4

Fix a classical globally hyperbolic spacetime with metric signature $(-+++)$; in this solution take $\hslash =1$. A free real Klein-Gordon field can be specified bywhere $\mu$ is its mass, $\xi$ its curvature coupling and $\mathcal{R}$ the scalar curvature. Global hyperbolicity ensures a well-posed initial-value problem on a Cauchy hypersurface $\Sigma$ and the existence of retarded and advanced propagators. Thus compactly supported field and normal-derivative data determine a classical solution. This fixes the dynamics, but not a Fock vacuum.

For real solutions with suitable support or falloff, the symplectic form on scalar-field solutions iswith $n$ the future unit normal. The field equation makes the current conserved, so this symplectic form is independent of $\Sigma$ when boundary flux vanishes. Quantize the initial data by the canonical commutation relation: with $\pi =n^a{\nabla }_a\varphi$ and the delta function defined relative to $d\Sigma$, $[\hat{\varphi }(x),\hat{\pi }(y)]=i{\delta }_{\Sigma }(x,y)$ and the two equal-field commutators vanish. Equivalently, construct the field algebra using the causal propagator. A state on that algebra is additional input.

To construct a particle representation, complexify the classical solution space. Its conserved Klein-Gordon inner product isIt is indefinite on all complex solutions. Choose a complete positive-norm subspace and an orthonormal mode basis $u_i$ with $(u_i,u_j)={\delta }_{ij}$, $(u_i^{\ast },u_j^{\ast })=-{\delta }_{ij}$ and $(u_i,u_j^{\ast })=0$. Such a choice is encoded by a compatible complex structure on the Klein-Gordon solution space. Its positive subspace gives the one-particle Hilbert space, and the associated bosonic Fock space contains symmetrized many-particle states. The field expansion isThe creation operator $a_i^{†}$ adds a particle in mode $i$, the annihilation operator removes one, and the number operator is $N_i=a_i^{†}{a}_i$. For continuous mode labels the sums and Kronecker deltas become integrals and delta functions, or one can work with normalized wave packets.

The ambiguity is precisely that the field equation and global hyperbolicity do not select that positive subspace. A different normalized basis may mix $u_j$ and $u_j^{\ast }$ by a Bogoliubov transformation, and then its annihilation operators mix $a_j$ and $a_j^{†}$. Its Fock vacuum and number operators differ. The Hadamard condition constrains physically acceptable short-distance singularities and allows local renormalization, but it still leaves many states. Hence there is generally no observer-independent particle count on an arbitrary dynamical geometry.

In a stable strictly stationary spacetime, a globally future timelike Killing vector field $K$ gives a preferred time translation. Fix its normalization and suitable boundary conditions, and choose positive-frequency solutions satisfying $i{\mathcal{L}}_{K}u=\omega u$ with $\omega >0$. The corresponding positive spectral subspace gives the preferred vacuum state in a stationary spacetime. Unitary changes of basis within it leave the vacuum and the particle notion unchanged. This construction assumes a well-defined positive stationary generator; stationarity by itself is insufficient if $K$ becomes spacelike, as in a Kerr ergoregion, or if unstable or zero modes obstruct the ground-state construction. It selects a preferred ground state under the stated assumptions, not a unique state among all thermal and excited states.

If the geometry is suitably stationary in the asymptotic past and future, choose those preferred mode spaces separately, giving the in-vacuum and out-vacuum. Propagate the past modes through the intervening region using the field equation and compare them to the future modes using the conserved Klein-Gordon inner product. Adopt the conventionThe canonical identities for a bosonic Bogoliubov transformation read $\alpha {\alpha }^{†}-\beta {\beta }^{†}=I$ and $\alpha {\beta }^T=\beta {\alpha }^T$. Extracting the future annihilation operator with the same inner product givesIn the in-vacuum, only $⟨a_j{a}_k^{†}⟩={\delta }_{jk}$ contributes to $⟨b_i^{†}{b}_i⟩$. Therefore the particle number from Bogoliubov coefficients isNonzero $\beta$ is the production of future particles from the past vacuum. Summing over future modes gives the total expected particle number when that sum is finite. For infinitely many modes, a Hilbert-Schmidt operator $\beta$ is the condition for unitary implementability between these pure bosonic Fock space representations; finite-volume or wave-packet calculations must respect the relevant measures and convergence. Particle production is determined by the negative-frequency mixing, rather than by identifying a single instantaneous vacuum throughout the time-dependent region.

The laws of black-hole mechanics initially relate geometric quantities in a way resembling thermodynamics. Hawking radiation supplies a physical temperature: in units $c=1$, while displaying $G,\hslash ,k_B$,The Hawking temperature $T_H$ is measured with the stationary time normalized at infinity. The radiation's thermal occupation factor is physically observable; propagation to infinity also introduces greybody factors, so the distant spectrum need not be a perfect blackbody spectrum at every frequency.

The Zeroth law of black-hole mechanics says that surface gravity $\kappa$ is constant on a stationary Killing horizon under its standard hypotheses. Through the Hawking temperature, this becomes uniform equilibrium temperature. The First law of black-hole mechanics isUsing $T_H$ identifies its area term as $T_H\delta S_{\mathrm{BH}}$. Integrating $\delta S_{\mathrm{BH}}=k_B\delta A/(4G\hslash )$ gives the Bekenstein-Hawking entropy, up to an additive constant. Thus $M$ is the energy, the angular and charge terms are work terms, and the geometrical law is the ordinary thermodynamic first law with a fixed entropy normalization.

The second law of black-hole mechanics, or Hawking's area theorem, gives nondecreasing area in the classical setting with the requisite energy and predictability assumptions. It corresponds to increasing Bekenstein-Hawking entropy. Semiclassical black-hole evaporation can decrease the area: the classical null energy condition need not hold for the quantum expectation of the stress-energy tensor. The appropriate extension is the generalized second law, that $S_{\mathrm{BH}}+S_{\mathrm{outside}}$ does not decrease, with the exterior entropy and its renormalization treated consistently. Hawking's temperature identification motivates this law; thermality alone does not prove every form of it.

The third law of black-hole mechanics is the unattainability, by an admissible finite physical process, of zero surface gravity. With Hawking temperature it becomes unattainability of absolute zero. It is not the assertion that an extremal black hole has vanishing entropy: its area can remain nonzero when $\kappa =0$.

For a Schwarzschild black hole, $\kappa =1/(4GM)$ and $A=16\pi G^2{M}^2$, giving $T_H=\hslash /(8\pi k_{B}GM)$ and $S_{\mathrm{BH}}=4\pi k_{B}G{M}^2/\hslash$. Their product satisfies $T_{H}d{S}_{\mathrm{BH}}=dM$, explicitly checking the first law. Quantum radiation turns the temperature and entropy in the mechanical analogy into physical thermodynamic quantities.